Question

The number of four letter words that can be formed using the letters of the word $BARRACK$ is :

• A. $120$
• B. $144$
• C. $264$
• D. $270$

Answer 1

The correct option is D $270$

BARRACK

No. of letters$=7$

$B\to 1$, $A\to 2$, $R\to 2$, $C\to 1$, $k\to 1$

No. of different letters$=5$

To make the $4$ letters words the possible cases:

$1)$ $2$ alike $2$ alike

$2)$ $2$ alike $2$ different

$3)$ all the $4$ alike are different.

Case I: $2$ alike and $2$ alike letters can be chosen out of three letters. (i.e., A, R)

${}^2{C}_2\times \frac{4!}{2!2!}=\frac{1\times 4\times 3\times 2}{2\times 2}=6$ ways.

Case II: $2$ alike, $2$ different can be chosen alike letters out of $2$ letters and $2$ different letters can be chosen out of remaining $4$ different letters. Thus, the number of words of such kind are ${}^2{C}_1\times {}^4{C}_2\times \frac{4!}{2!}=2\times 6\times 12=144$ ways.

Case III: All the $4$ letters are different. Out of $5$ different letters the number of ways of choosing $4$ letters $={}^5{C}_4=5$ ways

Thus, the number of words of such kind are $=5\times 4!=120$ ways

Thus, the total number of $4$ letter words which can be formed $=6+144+120=270$ ways.